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\magnification = 2000 

\input amstex
\documentstyle{amsppt}
\input OurATOMacros

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% File Name as ATO: z --> z^ee + ee z.tex

\cl{{ \bf Complex Map $ z\mapsto z^{ee} + ee\cdot z\ \  $ }{\footnote"*"{\verysmall 
This file is from the 3D-XplorMath project.  
Please see: \hfill\break \phantom{http://} http://3D-XplorMath.org/} } }

\cl{\bf (Default: $z\to z^2+2z$)}


\lf
Look at the functions $z\to z^2$, $ z\to 1/z$ and their ATOs first.

\lf
Of course, since $z^2 + 2z +1 = (z+1)^2$, this function is not very
different from the first example $z\to z^2$. But the change puts the
critical point to $-1$ on the unit circle ($f'(-1)=0$). Therefore, if
one looks what this map does to a Polar Grid, one can study the behaviour
near the critical point $z=-1$ with a different grid picture than in
the first example. Circles outside the unit circle are mapped to
Lima\c cons (Plane Curves Category) which wind around $-1$ twice. The
unit circle is mapped to a Cardioid and one can see the interior
angle of $180^\circ $ of the unit circle at $-1$ mapped to the
interior angle of $360^\circ$ of the Cardioid at $-1$. Also one can see
that a neigbourhood of $-1$ is strongly contracted by this function.
 \lf
See the function $z\to z+1/z$ next.

\ni
H.K.



\bye